5. LOP-SIDED MODES

Many galaxies have apparently lop-sided disks. The treatment here
will not go into detail, since
Jog & Combes
(2009)
have recently reviewed both the observational data and theory.

Both theoretical and simulation work on m = 1 distortions to an
axisymmetric disk require special care, since the absence of
rotational symmetry can lead to artifacts unless special attention is
paid to linear momentum conservation. Rigid mass components present
particular difficulties, since they should not be held fixed, and
extensive mass components are unlikely to respond as rigid objects.

As noted above,
Zang (1976)
found that the dominant instability of
the centrally cut-out Mestel disk was not the usual bar instability,
but a lop-sided mode, which persists in a full mass disk no matter how
large a degree of random motion or gentle the cutouts. This
surprising finding was confirmed and extended to general power-law
disks by
Evans & Read
(1998).
A lop-sided instability dominated simulations
(Sellwood 1985)
of a model having some resemblance to Zang's, in that
it had a dense massive bulge and no extended halo, while
Saha et al.
(2007)
reported similar behavior in simulations of a bare exponential disk.
Lovelace et al.
(1999)
found pervasive lop-sided instabilities near the disk
center in a study of the collective modes of a set of mass rings.

Various mechanisms have been proposed to account for this instability.
Baldwin et al.
(1980)
and
Earn &
Lynden-Bell (1996)
explored the idea that long-lived
lop-sidedness could be constructed from cooperative orbital responses
of the disk stars, along the lines discussed for bars by
Lynden-Bell (1979).
Tremaine (2005)
discussed a self-gravitating secular instability in
near-Keplerian potentials. 6
A more promising mechanism is a cavity mode, similar to that for the
bar-forming instability
(Dury et al.
2008):
the mechanism again supposes
feedback to the swing-amplifier, which is still vigorous for
m = 1 in a full-mass disk.

Feedback through the center cannot be prevented by an ILR for m =
1 waves, since the resonance condition
p =
-
R (eq. 5)
is satisfied only for retrograde waves.
But the lopsided mode can be stabilized by reducing the disk mass,
which reduces the X parameter (eq. 12) until amplification
dies for m = 1
(Toomre 1981).
Sellwood & Evans
(2001)
showed that, together with
a moderate bulge, the dark matter required for a globally stable disk
need not be much more than a constant density core to the minimum halo
needed for a flat outer rotation curve.

A qualitatively different lop-sided instability is driven by
counter-rotation. This second kind of m = 1 instability was first
reported by
Zang & Hohl (1978)
in a series of N-body simulations designed
to explore the suppression of the bar instability by reversing the
angular momenta of a fraction of the stars; they found that a
lop-sided instability was aggravated as more retrograde stars were
included in their attempts to subdue the bar mode. Analyses of
various disk models with retrograde stars
(Araki 1987,
Sawamura 1988,
Dury et al.
2008)
have revealed that the growth rates of lop-sided instabilities
increase as the fraction of retrograde stars increases.
Merritt &
Stiavelli (1990)
and
Sellwood & Valluri
(1997)
found lop-sided instabilities in simulations of oblate
spheroids with no net rotation. Their flatter models had velocity
ellipsoids with a strong tangential bias, whereas
Sellwood & Merritt
(1994)
found that disks with half the stars retrograde, together with moderate
radial motion were surprisingly stable.

Weinberg (1994)
pointed out that lop-sided distortions to near spherical
systems can decay very slowly, leading to a protracted period of
"sloshing". This seiche mode in a halo, which decays
particularly slowly in mildly concentrated spherical systems, could
provoke lop-sidedness in an embedded disk
(Kornreich et
al. 2002,
Ideta 2002).

If lop-sidedness is due to instability, then the limited work so far
suggests that it would imply a near-maximum disk. But lop-sidedness
in the outer parts could also be caused by tidal interactions, or
simply by asymmetric disk growth, with the effects of differential
rotation being mitigated perhaps by the cooperative orbital responses
discussed by
Earn &
Lynden-Bell (1996).